Dann im dunkeln Gebüsch und an der Seite des
Felsens

Schweben siehet ihr Bild, wohin er die Blicke nur
wendet,

Eilet es vor und glänzt und schwankt in herrlichen
Farben:

So bewegte vor Hermann die liebliche Bildung
des Mädchens

Sanft sich vorbei, und schien dem Pfad' ins
Getreide zu folgen.

10. Translate

Mit königlichen Gütern segne dich

Die Göttin! Sie gewähre Sieg und Ruhm
Und Reichthum und das Wohl der Deinigen
Und jedes frommen Wunsches Fülle dir!
Dass, der du über viele sorgend herrschest,
Du auch vor vielen seltnes Glück geniessest.

11. Translate—

Ja! Vieles reizt mich hier, ich will's nicht läugnen,
Mich reizt die bunte, kriegerische Bühne,
Die vielfach mir ein liebes Bild erneuert,
Mir an das Leben, an die Wahrheit knüpft,
Was mir ein schöner Traum nur hat geschienen.

12. Translate into German—

It has sometimes been remarked that Prussia, and more particularly Berlin, has an unfair share of knights, but the explanation given is no doubt true-namely, that Berlin often attracts later in life the best men in every branch of learning, in whatever part of Germany they may have been born. It has also been objected that hardly any one, however distinguished he may be, is elected before he is past fifty; but this, too, can hardly

be surprising, because it is only in recognition of original work and work of a permanent value, that these knights are chosen.

13. Translate into German

Where is she now? Call'd far away
By one she dared not disobey,
To those proud halls for youth unfit,
Where princes stand and judges sit.
Where Ganges rolls his widest wave
She dropped her blossom in the grave;
Her noble name she never changed
Nor was her nobler heart estranged.

14. Translate

Geist der Liebe, Weltenseele, Vaterohr, das keine
Stimme überhöret der dich lobenden Gemeinde!
Eine Reihe Dankgebetes, Lobgesangs ein Faden,
Zieht sich hin vom Duft des Morgens zu des
Abends Scheine

Lass in deinem Abendwinde Rosen säuseln über
Eines jeden, der dir sang, nun schlummernde
Gebeine!

Lass den freien Dichtermund hier deinem Lobe
dienen,

Bis in Engelzungen dort sich freier mischet seine!

ALGEBRA AND TRIGONOMETRY.

1. Resolve

Professor Nanson.

a (b − c) (b + c − a) + the two similar terms

into factors.

3. If x, y, z be positive quantities satisfying the

be in harmonical progression, prove that

27r29pqr + 2q3 = 0,

and solve the equation.

5. If the two series

Uo + U1 + U2 + &c.,

vo + v1 + V2 + &c.,

be convergent independently of the signs of the terms, prove that the series whose (n + 1)th term is

6. Prove that the coefficient of x+y+1 in the

7. If ƒ (x) + ƒ (y) = ƒ (xy), prove that

f(x) = f (a) log,

where a does not contain x.

8. If p and q be positive integers prove that

9. Prove that the a priori probability of making the same throw twice in succession with a pair of dice is s

Find the probability of doing the same with three dice.

10. If ABC, A'B'C' be two triangles such that

11. A person wishing to ascertain the height of a tower stations himself in a horizontal plane through the base at a point at which the elevation of the top is 30°. On walking a distance a in a certain direction, he finds the elevation of the top is the same as before, and on walking a distance five-thirds of a, at right angles to his previous direction, he finds that the elevation of the top is 60°. Shew that the height of the tower is a ora.

12. Prove that if in measuring the three sides of a triangle a small error x be made in the side a, the consequent error in the diameter of the circumscribing circle will be

x cosec A cot B cot C.

GEOMETRY AND TRIGONOMETRY.

Professor Nanson.

1. ABC is a triangle and O is any point within it. AO, BO, CO meet BC, CA, AB respectively in D, E, F. Prove that

2. Prove that the square of the tangent from any point on a given circle to a second given circle varies as the perpendicular from that point upon the radical axis of the two circles.

Find the locus of a point such that the tangents from it to two given circles may have a given

ratio.